Deeper than primes

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Apathia said:
I think it would remove a lot of mystery about OM, if we could bring in a 5 year old to demonstrate her grasp of counting numbers.
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Reminds me of the Groucho Marx quote:

"Why, a four-year-old child could understand this report. Run out and find me a four-year-old child. I can't make head nor tail out of it."
 
doronshadmi said:
What do you mean arbitrary?
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http://dictionary.reference.com/search?q=arbitrary

–adjective
1. subject to individual will or judgment without restriction; contingent solely upon one's discretion: an arbitrary decision.


2. decided by a judge or arbiter rather than by a law or statute.


3. having unlimited power; uncontrolled or unrestricted by law; despotic; tyrannical: an arbitrary government.


4. capricious; unreasonable; unsupported: an arbitrary demand for payment.


5. Mathematics. undetermined; not assigned a specific value: an arbitrary constant.
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doronshadmi said:
These are the only two states (uniquely chosen or not-uniquely chosen) that can be under this particular Persons\Glasses framework, so?
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Thus the selection of a particular glass by a particular person is arbitrary in your given “Persons\Glasses framework”

doronshadmi said:
The word "observer" has no meaning if the observed is not in the same framework with the observer. In other words, the property belongs to the framework and not to any particular relation or element under this framework.
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Observer to observed is that “particular relation” and it is the ascriptions of their meanings that sets your “framework” . The mere ascription of an observer sets at least the expectation of such a common framework. However, observer or an observation does not lose any meaning simply when there is nothing to observe. In fact such an observation, as in the observation of no voltage, is entirely plausible and a valid observation. So the framework need not include what is intended to, or the anything that can be, observed by a particular observer, yet the observation or the observer (like my voltmeter) still has meaning in and of itself.

doronshadmi said:
If the observer is totally isolated from the observed, then both words have no meaning.
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Words are given meaning by the common lexicon or language, they do not suddenly lose meaning when ones distinguish or isolates properties of the observer from the properties of the observed when the observer and the observed are distinguishable from each other. In fact that generally is the purpose of the words observer and observed to distinguish between those perhaps arbitrarily ascribed or even mutually exclusive perspectives. What happened to distinction being “a first order” property of your notions. You seem to be requiring the observer and observed to be indistinct which of course would make the possible observer / observed distinction, well, meaningless. That distinction itself is not inherently part of their meanings though, as I can be both the observer and the observed.


doronshadmi said:
What give them their meaning are the properties of the framework, where both of them are inseparable factors of it.
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They already have meaning and it is the ascription of their inherent meanings, arbitrary or not, that sets that framework. So as usual you have it backasswards they do not gain meaning from the framework, but set the framework and give that framework some particular meaning by their ascription or simply the ascription of an observer without something to observe.
 
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dlorde said:
I don't - I'm very aware of myself and my experience of understanding when doing research. I know that the results of my research are my interpretation of the data, and this is why I try to understand it from as many points of view as I can.
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The main idea is to use a language that does not reduce the many possible points of view of the researched into a one and only one agreed point of view about a given notion.

Standard Mathematics is exactly an education method that does its best in order to reduce any given notion into a one and only one agreed description (what is callad "a well-defined ..."). Any other view that does not agree with the already agreed definition, is automatically marked as nonsense or "out of Mathematics" gibberish, cranky etc. …
dlorde said:
I don't see how any research can be done without the researcher's awareness being present and inseparable from it - the researcher's awareness performs the research.
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I agree with you, but if you tell it to jsfisher or The Man they will try to convince you that the researcher is not a significant factor of any result (abstract or not).

Furthermore, they will tell you that any result where the researcher is involved, is not a valid result and should be ignored. They are using a naïve approach about the subjective influence of the researcher on the objective results. Their approach is naïve because their school of thought did not develop any comprehensive method that systematically researches the non-trivial interactions between the researcher and the researched, and this is exactly where Organic Mathematics gets on the scientific stage, and instead of eliminating the researcher as a subjective "white noise" (as Standard Math does for many years), it develops tools in order to research the possible interactions between the researched and the researcher, for at least to purposes:

1) To rigorously avoid subjective tendencies that may harm the validity of the results.

2) To rigorously research the researcher's objective properties (the properties that are not changed from person to person) and how the researcher's objective properties are related to the researched subject.

dlorde said:
What education methods are these? how are they used? Please give an example.
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To teach students that each one of us is like a cell of a one organism (branches of the same tree, if you like), such that at the basis of our different points of view and many levels of awareness, there is a common ground for both Ethics and Logics that can be developed into a one scientific method, which under its wings people will no longer use the language of Mathematics in order to develop destructive technologies, that their aims is to kill other people that get things (abstract or not) different than them.
 
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The Man said:
Words are given meaning by the common lexicon or language,
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And how that lexicon or language come into existence?

Who gives the meaning to words?

The Man said:
Thus the selection of a particular glass by a particular person is arbitrary in your given “Persons\Glasses framework”
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No, this framework of two distinctions is not less than both Parsons AND Glasses, and it is not limited to any particular choice of any glass by any particular person.

This model simply shows the all possible states of distinctions under the given framework.
The Man said:
However, observer or an observation does not lose any meaning simply when there is nothing to observe.
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Who knows that there is nothing to be observed?
 
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dlorde said:
This appears to be a deliberate attempt to integrate/amalgamate the observer and the observed into a complex and subjective relational 'entity'. Quite how this provides practical benefit is unclear, as is also how it avoids throwing out the baby of (scientific) objectivity with the bathwater.
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In post 4040 you wrote:
Well I get the idea, as Apathia explained it, and, as I already said, it seems to fit with what's in the document, but I would like to see a worked example of a real-world situation to which this language has been applied to analyse and/or resolve a mathematical/scientific problem with its moral and ethical implications.

Again: I would like to see a worked example of a real-world situation to which this language has been applied to bridge science and ethics.

It's surely not much to ask, as this is exactly the task the language is designed for?
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Let's keep insisting on concrete answers to these questions.
 
ddt said:
Yes, of course I used a computer. The large numbers involved pose all kind of interesting optimization questions, both w.r.t. time and space. I now have a generator for the list of partitions of n which runs in O(n) space :D. Your algorithm gives:

Or(100) = 2199516347439237513754580009632686701940043714513581388093

(okay, that one still took 5 hours of computer time).

I've seen the sequence at the OEIS. It seems to be the same - at least for the first 30 numbers of that sequence that are listed. Have you tried proving they are the same? Have you submitted your formula to the OEIS?


Technically, your definition of D:
D(alpha) = Prod (i=i..n) ...
still leads to an infinite recursion. If you restrict that one to
D(alpha) = Prod(i=1..n-1)
it doesn't matter if the sum in the definition of Or includes the partition n=n or not.

But that's a technical argument and not a conceptual one. There should be a rationale behind the formulae, and why the partition n=n is included or not. I'll have a look later at your other posts.
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Hi ddt,

Really a great work !
Thank you for the 5 hours of computing or(n).
You are the leader today of this computation on the planet of Earth..

Did you have in advance the program for the partitions ?

Best
Moshe:boxedin:
 
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The Man said:
It would seem to me then that what your are referring to as ‘distinction’ is simply an arbitrary choice made by the observer. As such would relate to the properties of the observer. In fact that such distinction is not inherently part of the object, the possible distinctions to select from is also something simply arbitrarily imposed by said observer. Thus as opposed to an observer not ignoring his relation with the observed you seem to be simply imposing additional arbitrary relationships that are not inherently part of the specific observation, but simply a matter of happenstance and arbitrary selection on the part of the observer. As the general purpose of observation is to determine characteristics about the observed, such imposed arbitrary aspects simply on the part of the observer detract from that. It would seem that if the focus was on the act of observation itself and specifically on the aspects the observer could bring to that observation that such consideration might have validity as opposed to simply detracting from a focus on what is observed. Generally conditions like control groups, double blind procedures and independent verification are used to isolate arbitrary aspects of the observer from the observed.
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Hi Man,

I think that you got the point !
best
Moshe
:boxedin:
 
MosheKlein said:
Thank you Little 10 Toes , for the nice link on Frege !
I think that during 2004 we discover the conection of OM
to his fundamental work in logic.

Best
Moshe :boxedin:
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Are you saying you had not seen axiomatic predicate logic before discovering Frege's place in Mathematics history in 2004?
 
Apathia said:
Yes, you're right, not permutations or iterations, but quite an increasing load of fract-thingies to bear in mind.

I think it would remove a lot of mystery about OM, if we could bring in a 5 year old to demonstrate her grasp of counting numbers.

Not County Cork.
I'm an American, born in the State of Florida.


Thank's
Too bad I don't speak Hebrew.
The situation speaks for itself though.
I wish you could set us down with the toy cars and the dots and show us in concrete the mathematic conceptions of pre-schoolers.
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Hi Apathia,

I will be glad to visit your place and demonstrate my work in mathematics with young childrens Do you have some good concetion with some kindergarden..in your area ?


Best
Moshe:boxedin:
 
jsfisher said:
I think we can agree, then, that Frege deserves some credit as the father of first order logic. However, that doesn't answer my question. I know what first-order means in Mathematics. I will assume you do, too. My question to you, MosheKlein, is how to express "distinction is a first-order property" (or relation or whatever) as a first-order predicate.

Such a predicate and a functional definition of distinction are necessary to understand what you mean by distinction in a more formal way.
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Hi jsfisher,

Basically I mean that Kolmogorov axioms on probability theory really were not relate to the revolution in QM during his time. Distinction is a new type of uncertainty whice fit the Quantume univers.


Best
Moshe:boxedin:
 
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MosheKlein said:
Hi jsfisher,

Basically I mean that Kolmogorov axioms on probability theory really were not relate to the revolution in QM during his time. Distinction is a new type of uncertainty whice fit the Quantume univers.
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This still doesn't address my question about your use of the term, first order.
 
jsfisher said:
This still doesn't address my question about your use of the term, first order.
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When you are really agree to the notion of distinction
( and I am not sure yet that you hade agree on that)
Then the true false logic is change immediately
For example : a line is belong and not belong to a point

Moshe:cool:
 
doronshadmi said:
And how that lexicon or language come into existence?
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From the desire and need to communicate. Some aspects come into existence arbitrarily while others directly relate to visual or auditory aspects they are intended to communicate, such as a pectoral or visual language and Onomonopia in spoken language.


doronshadmi said:
Who gives the meaning to words?
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If you are referring to definitions of words that is simply common usage. However you tend to give your own personal meaning to words that often go directly against such common usage. As I stated above some words or representative aspects of language are not “given” but in fact taken from what they are intended to represent.


doronshadmi said:
No, this framework of two distinctions is not less than both Parsons AND Glasses, and it is not limited to any particular choice of any glass by any particular person.
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That’s what makes the choice arbitrary in that framework. Can the person choose some glass other then the two on the table? Can some other person choose a glass on that table. If not then the choices of glasses and people are in fact extremely limited that limitation is apparently as arbitrary as the required choice with in the limited framwork.

doronshadmi said:
This model simply shows the all possible states of distinctions under the given framework.
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Since no framework is established to provide those distinction, such distinctions are again simply arbitrary. Entirely at the discretion of the chooser. As the second person observing the chooser select a glass how is that arbitrary distinction and selection required in any way to be altered or determined by my observation?

doronshadmi said:
Who knows that there is nothing to be observed?
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Who knows that there is something to be observer if it is not observed? The observed requires an observer, the observer is not necessarily required to actually observe anything. If I close my eyes and do not observe the chooser selecting a glass does that leave my glass still indistinct and a selection to be made by me simply because I did not observe the selection? If again I close my eyes and mentally select the glass on my left, but the chooser takes the glass to my left then rotates the table so that the remaining glass is to my left do I still have the glass on my left I mentally selected when I open my eyes?
 
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MosheKlein said:
Hi Apathia,

I will be glad to visit your place and demonstrate my work in mathematics with young childrens Do you have some good concetion with some kindergarden..in your area ?


Best
Moshe:boxedin:
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Generous of you, Moshe, but I have no relation with a local kindergarten which would welcome your demonstration.

So perhaps you could share a typical interaction via a simple dialog.
For example what I did in the Paddy Post.
You describe your set up then give Paddy's response to each situation.

That way you could illustrate the typical way a pre-schooler thinks and how that relates to OM.

If you could do this, it would certainly be more economic and less time consuming than you flying to America. Also you'd be able to cut to the chase rather than waiting for little kids to make the demonstration you expect.
 
doronshadmi said:
The main idea is to use a language that does not reduce the many possible points of view of the researched into a one and only one agreed point of view about a given notion.
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You seem to be implying that most people have a very one-dimensional view of the world, and need a special language to bring ethics and morals into their consideration. I can assure you that there are a lot of people out there who recognise the many points of view about the issues they deal with, and the moral and ethical implications, without needing such a tool. You have yet to supply any example that shows how your language helps achieve this, and any practical example at all of how it works in practice would be welcome.

I agree with you, but if you tell it to jsfisher or The Man they will try to convince you that the researcher is not a significant factor of any result (abstract or not).
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The significance of the researcher depends on the domain under consideration. Science has had its successes by a philosophy and methodology that attempts to minimise the influence of extraneous influences, including the researcher, on the results of the study. This does not mean that these influences are necessarily neglected or are of no interest, but unless the intention is to study them, they are outside the domain of the study (by definition). Of course, the framework within which science operates is guided by moral and ethical principles, and these decide the moral and ethical parameters of the work done and how the results are used. If your point is that there is insufficient moral and ethical influence (or control), then I'd agree - up to a point. But science supplies the evidence on which moral and ethical decisions can be made, whether by politicians, regulators, or the scientists themselves, and to do this it must supply 'clean' (unbiased and objective) data. I still await some example of how your language enables the moral and ethical considerations to be bundled into the scientific method or the raw science.

Furthermore, they will tell you that any result where the researcher is involved, is not a valid result and should be ignored. They are using a naïve approach about the subjective influence of the researcher on the objective results.
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Again, this depends on the domain under consideration. There has been plenty of work that examines the influence and interaction of the researcher with both experimental process and the results. It's ironic that the tedious and expensive double-blind protocol emerged from these studies - the research into the interaction between researcher, experiment, and results identified the contexts in which science is not effective unless the researcher's potential bias is controlled.
... Organic Mathematics ... develops tools in order to research the possible interactions between the researched and the researcher, for at least to purposes:

1) To rigorously avoid subjective tendencies that may harm the validity of the results.

2) To rigorously research the researcher's objective properties (the properties that are not changed from person to person) and how the researcher's objective properties are related to the researched subject.
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As I already said, there has been a lot of research in these areas and DBTs are one solution that has emerged to your point 1. I think your point 2 is related to psychology, but it's not clear quite what you mean by the 'properties that are not changed from person to person'.

To teach students that ... there is a common ground for both Ethics and Logics that can be developed into a one scientific method, which under its wings people will no longer use the language of Mathematics in order to develop destructive technologies, that their aims is to kill other people that get things (abstract or not) different than them.
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You do seem to have a hangup about the use of maths/science to develop destructive technologies. You can teach kids what you like, but nuclear technology can supply us with electrical power or with political power, and the decision about the way it will be exploited is generally not within the scope of the researchers.

I'm trying to grasp how your ideas will have any practical effect, and I still can't see it. If you can't give a real-world example of how it might work, invent a hypothetical one so I can see the moral and ethical dimensions are incorporated into the science using your language.
 
MosheKlein said:
When you are really agree to the notion of distinction
( and I am not sure yet that you hade agree on that)
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I am still hoping you will be willing and able to explain what you mean by the term, distinction.

...Then the true false logic is change immediately
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No, it doesn't. You may have developed a legitimate alternative logic, but the basic one sitting at the foundations of Mathematics doesn't just change in a puff of distinction.
 
The Man said:
Who knows that there is something to be observer if it is not observed? The observed requires an observer, the observer is not necessarily required to actually observe anything. If I close my eyes and do not observe the chooser selecting a glass does that leave my glass still indistinct and a selection to be made by me simply because I did not observe the selection? If again I close my eyes and mentally select the glass on my left, but the chooser takes the glass to my left then rotates the table so that the remaining glass is to my left do I still have the glass on my left I mentally selected when I open my eyes?
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OK The Man, I get your point, you simply wish not to get the Symmetric\Non-symmetric model and instead you rambling around the Person's personal life, his health problems, his relations with his parents, etc. etc. …

My suggestion to you is: go have a life …
 
dlorde said:
I'm trying to grasp how your ideas will have any practical effect, and I still can't see it. If you can't give a real-world example of how it might work, invent a hypothetical one so I can see the moral and ethical dimensions are incorporated into the science using your language.
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The current scientific paradigm, which easily ignores the real-time presence of the researcher's awareness during his scientific work (you call this ignorance "context dependent"), is exactly the method that artificially causes the dichotomy between Ethics and Logics.

It is a fundamental mistake to think that Ethical considerations cannot be an inseparable property of the scientific real-time tools of the exact sciences, and OM's goal is to develop exactly such tools that will give the scientist the needed Ethical aspects directly through his scientific tools.

If such a goal is achieved, then we do not need politicians or religious leaders in order to decide what to do with a destructive technology, simply because such a technology will not be developed by scientists that are aware of Ethics and Logics\Technologic aspects as built-in properties of their scientific tools.

You do not grasp yet the notion of the Organic Numbers and how they are used to train the mind to be aware of the fine relations between itself and its researched subjects, such that the researcher is aware of itself as an inseparable organ of a one realm (abstract or not) where the researcher is an important factor of the development of that realm.

The gate to the Organic Numbers realm is in pages 18-20 of http://www.geocities.com/complementarytheory/OMPT.pdf but somehow you miss it (you did not get yet the notion of the Cybernetic Kernels and Complexity development, which leads to a natural responsible researcher).
 
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The Man said:
Who knows that there is something to be observer if it is not observed? The observed requires an observer, the observer is not necessarily required to actually observe anything. If I close my eyes and do not observe the chooser selecting a glass does that leave my glass still indistinct and a selection to be made by me simply because I did not observe the selection? If again I close my eyes and mentally select the glass on my left, but the chooser takes the glass to my left then rotates the table so that the remaining glass is to my left do I still have the glass on my left I mentally selected when I open my eyes?
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At the risk of draggin the HPC debate into this thread [please forgive me >_<] I'd say that to be conscious is to observe something. Cutting of a particular sensory channel wouldn't change this. Even if one has their optic nerves severed, is rendered deaf, and loses connection to their bodily sensations, if they are still have thoughts, emotions, and mental imagery they are still observing.
 
doronshadmi said:
OK The Man, I get your point, you simply wish not to get the Symmetric\Non-symmetric model and instead you rambling around the Person's personal life, his health problems, his relations with his parents, etc. etc. …

My suggestion to you is: go have a life …
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I don't think he was rambling about someone's personal life. He was just conducting a thought experiment, is all :)
 
Ok, now having waded through the Sweden presentation videos, I think I understand what MosheKlein means by distinction. Let me play this back to MosheKlein for his reaction:

First off, the concept of number being used here has some underlying reference to things, often implicit. You don't just consider 3 as an abstract mathematical construct; there must be 3 things of one sort or another. Beads on a string has been the frequent example in this thread.

Each thing is presumed to have some identity, but the identity may be unknown to use. Distinction refers to the possible ways in which we can distinguish (or not) among those things.

Let's say we have 2 things, with identities A and B. If we don't know which is which, the best we can say is one is either A or B and the other is either B or A. (This would be the superposition of identities that's been mentioned once or twice in this thread.) If we know the identity of one of the things, then by the process of elimination we know the identity of the other.

So, for 2 things, there are 2 distinctions: (AB, AB) and (A, B).

For 3 things, the claim is there are 3 distinctions, ranging from knowing nothing, something, or everything: (ABC, ABC, ABC), (A, BC, BC), and (A, B, C).


How am I doing, MosheKlein?
 
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jsfisher said:
Ok, now having waded through the Sweden presentation videos, I think I understand what MosheKlein means by distinction. Let me play this back to MosheKlein for his reaction:

First off, the concept of number being used here has some underlying reference to things, often implicit. You don't just consider 3 as an abstract mathematical construct; there must be 3 things of one sort or another. Beads on a string has been the frequent example in this thread.

Each thing is presumed to have some identity, but the identity may be unknown to use. Distinction refers to the possible ways in which we can distinguish (or not) among those things.

Let's say we have 2 things, with identities A and B. If we don't know which is which, the best we can say is one is either A or B and the other is either B or A. (This would be the superposition of identities that's been mentioned once or twice in this thread.) If we know the identity of one of the things, then by the process of elimination we know the identity of the other.

So, for 2 things, there are 2 distinctions: (AB, AB) and (A, B).

For 3 things, the claim is there are 3 distinctions, ranging from knowing nothing, something, or everything: (ABC, ABC, ABC), (A, BC, BC), and (A, B, C).


How am I doing, MosheKlein?
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You are doing great.

The case of 1 is:

(A)

The case of 2 is:

(AB, AB)
(A, B)

The case of 3 is:

(ABC, ABC, ABC)
(AB, AB, C)
(A, B, C)

The case of 4 is:

(ABCD, ABCD, ABCD, ABCD)
(AB, AB, ABCD, ABCD)
(A, B, ABCD, ABCD)
(AB, AB, AB, AB)
(A, B, AB, AB)
(A, B ,A ,B)
(ABC, ABC, ABC, D)
(AB, AB, C, D)
(A, B, C, D)

Please write case 5.
 
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doronshadmi said:
The case of 4 is:

(ABCD, ABCD, ABCD, ABCD)
(AB, AB, ABCD, ABCD)
(A, B, ABCD, ABCD)
(AB, AB, AB, AB)
(A, B, AB, AB)
(A, B ,A ,B)
(ABC, ABC, ABC, D)
(AB, AB, C, D)
(A, B, C, D)
Click to expand...


You might want to reconsider a few of those.
 
jsfisher said:
As a mathematical term, property is usually restricted to yes/no propositions. This may seem like an unnecessary nit-pick, especially since I did say it was only usually true, but the phrasing first-order property puts it into the realm of always true.
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Alright. Thanks for the info :)
 
jsfisher said:
You might want to reconsider a few of those.
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The case of 4 is:

(1+1+1+1)
(ABCD, ABCD, ABCD, ABCD)

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

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

(3+1)
(ABC, ABC, ABC, D)
(AB, AB, C, D)
(A, B, C, D)
 
doronshadmi said:
...
(A, B, ABCD, ABCD)
...
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Taking just one as an example, you don't see any problem with this one? The example (A, B, ABCD, ABCD) expresses the situation where we know the identity of one thing, and the identity of another thing, but uncertain about either of the last two.

Without too much thought, most of us here could narrow the field a bit for those last two.
 
jsfisher said:
Taking just one as an example, you don't see any problem with this one? The example (A, B, ABCD, ABCD) expresses the situation where we know the identity of one thing, and the identity of another thing, but uncertain about either of the last two.

Without too much thought, most of us here could narrow the field a bit for those last two.
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In (A, B, ABCD, ABCD) the A, B part is a recursion of level 2 distinction within level 4 distinction, which has no influence on level 4 distinction.

Please look at this diagram for better understanding:

1-4.jpg


As you see, as long as the partition of 4 is not (3+1), we are under splitting\connecting transitions between partitions, and we can't determine for sure a unique id.

EDIT: When we at (3+1), then +1 has a stable and unique id.

Also, by using the diagram above we can represent case 4 in a more general way:

Instead of ABCD we use 4 in order to represent 4 possible ids.
Instead of ABC we use 3 in order to represent 3 possible ids.
Instead of AB we use 2 in order to represent 2 possible ids.
Instead of A we use 1 in order to represent 1 possible ids.

By using this generalization, case 4 looks like this:

(1+1+1+1)
(4, 4, 4, 4)

(2+1+1)
((2, 2), 4, 4)
(((1), 1), 4, 4)

(2+2)
((2, 2), (2, 2))
(((1), 1), (2, 2))
(((1), 1), ((1), 1))

(3+1)
((3, 3, 3), 1)
(((2, 2), 1), 1)
((((1), 1), 1), 1)
 
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doronshadmi said:
In (A, B, ABCD, ABCD) the A, B part is a recursion of level 2 distinction within level 4 distinction, which has no influence on level 4 distinction.
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You are telling how you got it. I don't care how you calculated it. I only care about what it means at this point.
 
jsfisher said:
You are telling how you got it. I don't care how you calculated it. I only care about what it means at this point.
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(A, B, ABCD, ABCD) means: the distinct from of 2 within form 4 ( it is under the partition (2+1+1) ).
 
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doronshadmi said:
In (A, B, ABCD, ABCD) the A, B part is a recursion of level 2 distinction within level 4 distinction, which has no influence on level 4 distinction.
Click to expand...

By the way, if you want to focus on your calculation method, you need to explain all the inconsistencies. For example, 4 = 3+1 should be (ABC, ABC, ABC, ABCD) to be consistent with your other "recursions." Why isn't it?
 
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